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String Principal Bundles and Courant Algebroids

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String Principal Bundles and Courant Algebroids String principal bundles and Courant algebroids Yunhe Sheng, Xiaomeng Xu and Chenchang Zhu Abstract Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids encode the infinitesimal symmetries of S1-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non- commutative analogue of exact Courant algebroids. In this article, we explore what the “principal bundles” behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and Chen-Stiénon-Xu on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the “higher Atiyah algebroid” loses some information and at the same time, only some special transitive Courant algebroids come from string bundles. Contents 1 Introduction 2 2 Preliminaries on prestacks, stacks and the plus construction 4 3 (3, 1)-sheaf of string principal bundles 6 3.1 Finite dimensional model of Stringp(G) .............................6 p+ 3.2 (3, 1)-sheaf BString(n)c of string principal bundles with connection data . .8 3.3 Lifting Theorem and Comparison . 12 p+ 4 (2, 1)-sheaf TCc of transitive Courant algebroids with connections 17 5 Morphism from the string sheaf to the transitive Courant sheaf 24 p+ p+ 5.1 Construction of the morphism Φ: BString(n)c → TCc ................... 25 p+ p+ 5.2 Property of the morphism Φ: BString(n)c → TCc ..................... 29 A Appendix 31 p+ A.1 (2, 1)-sheaf TLc of transitive Lie algebroids with connections . 31 p+ A.2 (2, 1)-sheaf ECc of exact Courant algebroids with connections . 33 A.3 Gluing transitive Courant algebroids via local data . 35 A.4 Inner automorphisms of transitive Courant algebroids . 37 0Keyword:Courant algebroids, string principal bundles, Pontryagin classes 0MSC: 18B40, 18D35, 46M20, 53D17, 58H05. 1 1 Introduction Just like Atiyah Lie algebroids encode the infinitesimal symmetries of G-principal bundles [3, 31], exact Courant algebroids are believed to encode the infinitesimal symmetries of U(1)-gerbes (or equivalently BU(1)-2-principal bundles) [10, 21, 35, 42]. Then it is proven later in [18] that the infinitesimal symmetries of an S1-gerbe equipped with a connective structure are precisely encoded by an exact Courant algebroid with a connection. Transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the “principal bundles” behind transitive Courant algebroids are. First, we notice that there are topological obstructions for the existence of such transitive Courant algebroids. In [9], Bressler discovered that the obstruction to extend an Atiyah Lie algebroid to a transitive Courant algebroid is given by the real first Pontryagin class. This is further fully generalized to the case of any regular Courant algebroid by Chen-Stiénon-Xu [17] in a differential geometry setting, where the authors gave a complete classification result. In fact, Ševera outlined some very nice ideas to classify transitive Courant algebroids in a series of private letter exchanges with Weinstein [42]. The role of the Pontryagin class is further developed in his later works [43, 44]. We then notice that the first Pontryagin class arises as an obstruction in another domain. Motivated by Stolz-Teichner’s program of topological modular forms [48], Redden [34] defined a string class on a Spin(n)-principal bundle P¯ → M, as a class ξ ∈ H3(P,¯ Z), such that for every point p ∈ P¯ the associated ¯ 3 inclusion ip : Spin(n) → P by g 7→ g · p pulls back ξ to the standard generator of H (Spin(n), Z). He further proved that the obstruction for a Spin(n)-principal bundle P¯ over M to admit string classes is 1 ¯ 4 provided by the integer class of half the first Pontryagin class 2 p1(P ) ∈ H (M, Z). In [50], Waldorf proved that such string classes on P¯ are in one-to-one correspondence with the isomorphism classes of trivializations of the Chern-Simons 2-gerbe over P¯, where the Chern-Simons 2-gerbe is again characterised 1 ¯ by 2 p1(P ). Topologically, Stolz and Teichner view the above string structure on a Spin(n)-principal bundle ¯ M −→P BSpin(n) as a lift of the structure group of P¯ from Spin(n) to a certain three-connected extension, the string group String(n), BString(n) : P¯ M / BSpin(n). 1 Thus, if we realise “String(n)-principal bundles” in differential geometry, one may interpret 2 p1 as the lifting obstruction of a Spin(n)-principal bundle to a String(n)-principal bundle. A more familiar fact in this style is that the lifting obstruction of a SO(n)-principal bundle M −→P BSO(n) to a Spin(n)-principal P˜ bundle M −→ BSpin(n) is given by w2(P ). In fact, there is a whole sequence, called the Whitehead tower: · · · → BString(n) → BSpin(n) → BSO(n) → BO(n), 1 with obstruction w1, w2 and 2 p1 respectively. Here w1 and w2 are the first and second Stiefel-Whitney classes. Since the topological obstruction for both transitive Courant algebroids and String(n)-principal bun- dles is provided by the first Pontryagin class, we naturally believe that the principal bundle behind a transitive Courant algebroid is exactly a String(n)-principal bundle. This belief is also supported by another observation from T-duality. Let us start with two T-dual torus bundles X, Xˆ over M, and matching T-dual S1-gerbes G → X and Gˆ → Xˆ, Bouwknegt-Evslin-Mathai [7] and Bunke-Schick [12] proved that the twisted K-theory for the T-dual pairs are isomorphic, that is, there is an isomorphism between twisted K-groups K•(X, G) =∼ K•(X,ˆ Gˆ). On the level of differential geometric objects, Cavalcanti-Gualtieri [16] proved that the exact Courant algebroid associated to the T- dual S1-gerbes are the same. Now we extend this story Spin(n)-equivariantly. We begin with two T-dual torus bundles X, Xˆ over M, and their matching T-dual string structures (P, ξ) → X and (P,ˆ ξˆ) → Xˆ, 2 where P and Pˆ are Spin(n)-principal bundles over X and Xˆ respectively, and ξ, ξˆ are string classes on P and Pˆ respectively. Leaving alone what the cohomological invariants should be, Baraglia and Hekmati [5] showed that, on the level of differential geometric objects, the transitive Courant algebroids associated to both sides are isomorphic. In this article, we realise String(n)-principal bundles and their connections as differential geometric p+ objects by describing the entire (3, 1)-sheaf (or 2-stack) BString(n)c . Then we make the connection between transitive Courant algebroids and string principal bundles explicit and functorial by constructing a morphism between their corresponding stacks. For this purpose, first we study what a String(n)-principal bundle with connection data really is. As we have seen, String(n) is a three-connected cover of Spin(n), and this forces the model of String(n) to be either infinite-dimensional or finite-dimensional however higher (namely being a Lie 2-group)1. We take the second approach with the model of Schommer-Pries [39] for String(n). The advantage of this model is that the spaces it involves are all nice finite dimensional manifolds, thus there is no additional analytic difficulty when solving equations or constructing covers; at the same time, this is paid off by algebraic difficulty of chasing through various pages of spectral sequences of cohomological calculation. p First we construct a (3, 1)-presheaf of String(n)-principal bundles with connection data BString(n)c p+ and complete it into a (3, 1)-sheaf (or a 2-stack) BString(n)c using the plus construction. This is essentially to build a String(n)-principal bundle with a connection from local data and gluing conditions in the fashion of Breen-Messing. Breen and Messing studied connections for gerbes in their original work [8]. See [15] for more details. We also notice that in a recent work [51], connections for 2-principal bundles of strict 2-groups are studied both locally and globally. However, the finite-dimensional differential geometric model for String(n) is a non-strict Lie 2-group. This forces us to develop our own formula instead of using existing results in literature. We also noticed that [22] develops a very nice model which explains connection data for all L∞-algebras based on the infinite dimensional integration model of L∞- algebras in [26, 28]. However, to relate their infinite dimensional model to our formula of describing connection with descent equations (Eqs. (5)-(8)), one needs to solve variables on ∆n in their model in some Lie type of PDE’s. This is a non-trivial task and we delay it to a future project with more specific focus. To justify our construction, we prove directly the lifting theorem that one expects for String(n)- principal bundles and provide a comparison to previous string concepts of Stolz-Teichner, Redden and Waldorf respectively in Section 3.3: P¯ Theorem 1.1.
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